Özet:
In this thesis, a new implicit-explicit local differential transform method (IELDTM)
with an arbitrary order is produced for stiff differential equations. In the produced
method, which is a stability-preserved numerical algorithm that accepts the idea of
differential transformation as a starting point, all information about the numerical
process is taken directly from the corresponding differential equations. The currently
proposed method is derived for stiff initial value problems (IVPs), stiff boundary value
problems (BVPs), and stiff parabolic initial-boundary value problems. All theoretical
analyses including priori error estimations and stability analysis are provided and the
theoretical order expectations are verified through various numerical experiments.
The IELDTM is found to be a high order, stability preserved, and versatile numerical
approach for both ordinary differential equations (ODEs) and parabolic partial
differential equations (PDEs) up to three spatial dimensions. The IELDTM is proven to
eliminate both some existing drawbacks of the differential transform-based methods
and other numerical techniques such as the finite element method (FEM) and finite
difference method (FDM) by providing optimized degrees of freedom.
